Renormalized and entropy solutions to the general nonlinear parabolic equations in Musielak-Orlicz spaces

TL;DR

This paper establishes the existence, uniqueness, and equivalence of renormalized and entropy solutions for general nonlinear parabolic equations in Musielak-Orlicz spaces, covering various growth conditions.

Contribution

It introduces a unified framework for well-posedness of nonlinear parabolic equations in Musielak-Orlicz spaces, including cases with Orlicz, variable exponent, and double-phase growth.

Findings

Existence and uniqueness of solutions are proven.

Renormalized and entropy solutions are shown to be equivalent.

Results apply to a broad class of growth conditions.

Abstract

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak-Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents, and double-phase growth.